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If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable, then its derivative $f'$ is Baire 1 (which essentially follows by the definition of derivative).

Do functions differentiable almost everywhere exhibit a similar kind of regularity? In particular, let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable almost everywhere and let $g(x)$ be $f'(x)$ if the latter exists at $x$, and zero everywhere else. Which function class does $g$ belong to?

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    $\begingroup$ Not an answer but still worth noting is the following result: A. Bruckner, Differentiation of Real Functions (2d ed, 1994), chapter 8 (“Derivatives a.e. and generalizations”), theorem 1.3 (rephrased to align with your notation): a necessary and sufficient condition on a function $g\colon [a,b]\to\mathbb{R}$ for there to exists a function $f$ such that $f' = g$ almost everywhere is that $f$ be measurable and finite almost everywhere; furthermore, when this condition is satisfied, $f$ can be chosen continuous. $\endgroup$
    – Gro-Tsen
    Commented Jun 14, 2023 at 19:54
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    $\begingroup$ (The difference between the result in my previous comment and your question is that you impose $g(x)=0$ when $f'(x)$ does not exist, whereas the above result simply requires $f' = g$ almost everywhere. But I still think it's interesting enough to mention.) $\endgroup$
    – Gro-Tsen
    Commented Jun 14, 2023 at 19:55
  • $\begingroup$ @Gro-Tsen Thanks for the answer, which is helpful: I am also interested in the behaviour of functions you mention ($f'=g$ ae), i.e. the what happens on the measure zero set is not that important. $\endgroup$
    – Sam Sanders
    Commented Jun 15, 2023 at 7:08

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